We investigate the magnetic dichotomy between Ap/Bp and other A-type stars by carrying out a deep spectropolarimetric study of Am and HgMn stars. Using the NARVAL spectropolarimeter at the Telescope Bernard Lyot (Observatoire du Pic du Midi, France), we obtained high-resolution circular polarisation spectroscopy of 12 Am stars and 3 HgMn stars. Using Least Squares Deconvolution (LSD), no magnetic field is detected in any of the 15 observed stars. Uncertaintiies as low as 0.3 G (respectively 1 G) have been reached for surface-averaged longitudinal magnetic field measurements for Am (respectively HgMn) stars. Associated with the results obtained previously for Ap/Bp stars, our study confirms the existence of a magnetic dichotomy among A-type stars. Our data demonstrate that there is at least one order of magnitude difference in field strength between Zeeman detected stars (Ap/Bp stars) and non Zeeman detected stars (Am and HgMn stars). This result confirms that the spectroscopically-defined Ap/Bp stars are the only A-type stars harbouring detectable large-scale surface magnetic fields.